On metric diophantine approximation in matrices and Lie groups

TL;DR

This paper investigates the diophantine approximation properties of submanifolds in matrix spaces and Lie groups, providing formulas and obstructions for extremality, with applications to rational nilpotent Lie groups.

Contribution

It identifies algebraic obstructions to extremality and derives formulas for diophantine exponents of algebraic submanifolds and rational nilpotent Lie groups.

Findings

Identified algebraic obstructions to extremality.

Derived formulas for diophantine exponents of algebraic submanifolds.

Applied results to rational nilpotent Lie groups.

Abstract

We study the diophantine exponent of analytic submanifolds of the space of m by n real matrices, answering questions of Beresnevich, Kleinbock and Margulis. We identify a family of algebraic obstructions to the extremality of such a submanifold, and give a formula for the exponent when the submanifold is algebraic and defined over the rationals. We then apply these results to the determination of the diophantine exponent of rational nilpotent Lie groups.